Irodov – Problems in General Physics

(Joyce) #1

  • Relationship between the energy and momentum of a relativistic par-
    tide
    E 2 — p 2 c 2 = mic 4 , pc = YT (T ±2moc 4 ).^ (1.8h)

  • When considering the collisions of particles it helps to use the follow-
    ing invariant quantity:
    E2 p2c2 = mic4,^ (1.81)


where E and p are the total energy and momentum of the system prior to the
collision, and mo is the rest mass of the particle (or the system) formed.

1.340. A rod moves lengthwise with a constant velocity v relative
to the inertial reference frame K. At what value of v will the length
of the rod in this frame be 1 = 0.5% less than its proper length?
1.341. In a triangle the proper length of each side equals a. Find
the perimeter of this triangle in the reference frame moving relative
to it with a constant velocity V along one of its
(a) bisectors; (b) sides.
Investigate the results obtained at V <c and V -4- c, where c is the
velocity of light.
1.342. Find the proper length of a rod if in the laboratory frame
of reference its velocity is v = c/2, the length 1 = 1.00 m, and the
angle between the rod and its direction of motion is 0 = 45°.
1.343. A stationary upright cone has a taper angle 0 =. 45°,
and the area of the lateral surface So = 4.0 m 2. Find: (a) its
taper angle; (b) its lateral surface area, in the reference frame
moving with a velocity v = (4/5)c along the axis of the cone.
1.344. With what velocity (relative to the reference frame K) did
the clock move, if during the time interval t = 5.0 s, measured by
the clock of the frame K, it became slow by At = 0.10 s?
1.345. A rod flies with constant velocity past a mark which is
stationary in the reference frame K. In the frame K it takes At
20 ns for the rod to fly past the mark. In the reference frame fixed
to the rod the mark moves past the rod for At' = 25 ns. Find the prop-
er length of the rod.
1.346. The proper lifetime of an unstable particle is equal to
Ato = 10 ns. Find the distance this particle will traverse till its
decay in the laboratory fraine of reference, where its lifetime is equal
to At = 20 ns.
1.347. In the reference frame K a muon moving with a velocity
v = 0.990c travelled a distance 1 = 3.0 km from its birthplace to
the point where it decayed. Find:
(a) the proper lifetime of this muon;
(b) the distance travelled by the muon in the frame K "from the
muon's standpoint".
1.348. Two particles moving in a laboratory frame of reference
along the same straight line with the same velocity v = (314)c strike
against a stationary target with the time interval At = 50 ns. Find

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